Generic vanishing fails for singular varieties and in characteristic p > 0

Abstract We give counterexamples to the degeneration of the Hochschild-Kostant-Rosenberg spectral sequence in characteristic p, both in the untwisted and twisted settings. We also prove that the de Rham-HP and crystalline-TP spectral sequences need not degenerate.

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Effect of oxygen vacancies and crystal symmetry on piezocatalytic properties of Bi2WO6 ferroelectric nanosheets for wastewater decontamination

Environmental Science Nano ◽

10.1039/d1en00022e ◽

2021 ◽

Author(s):

Zihan Kang ◽

Enzhu Lin ◽

Ni Qin ◽

Jiang Wu ◽

Baowei Yuan ◽

...

Keyword(s):

Organic Pollutants ◽

Oxygen Vacancies ◽

Mechanical Energy ◽

Crystal Symmetry ◽

Characteristic P ◽

Novel Technique ◽

Effect Of Oxygen

Piezocatalysis emerged as a novel technique to make use of mechanical energy in dealing with organic pollutants in wastewater. In this work, the ferroelectric Bi2WO6 (BWO) nanosheets with a characteristic...

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Topology of Subvarieties of Complex Semi-abelian Varieties

International Mathematics Research Notices ◽

10.1093/imrn/rnaa242 ◽

2020 ◽

Author(s):

Yongqiang Liu ◽

Laurentiu Maxim ◽

Botong Wang

Keyword(s):

Euler Characteristic ◽

Morse Theory ◽

Characteristic Property ◽

Abelian Varieties ◽

Betti Numbers ◽

Local Systems ◽

Affine Manifolds ◽

Cyclic Covers ◽

Rank One ◽

Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

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SUBALGEBRAS OF THE POLYNOMIAL ALGEBRA IN POSITIVE CHARACTERISTIC AND THE JACOBIAN

Journal of Algebra and Its Applications ◽

10.1142/s0219498811004756 ◽

2011 ◽

Vol 10 (04) ◽

pp. 605-613

Author(s):

ALEXEY V. GAVRILOV

Keyword(s):

Principal Ideal ◽

Positive Characteristic ◽

Polynomial Algebra ◽

Characteristic P ◽

The Ideal

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x1, …, xn]. Let J(R) be the ideal in 𝕜[X] defined by [Formula: see text]. It is shown that if it is a principal ideal then [Formula: see text], where q = pn(p - 1)/2.

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Geometry and topology of the space of Kähler metrics on singular varieties

Compositio Mathematica ◽

10.1112/s0010437x18007170 ◽

2018 ◽

Vol 154 (8) ◽

pp. 1593-1632 ◽

Cited By ~ 3

Author(s):

Eleonora Di Nezza ◽

Vincent Guedj

Keyword(s):

Einstein Metrics ◽

Normal Space ◽

Fano Varieties ◽

Kähler Metrics ◽

Kahler Metrics ◽

Metric Properties ◽

Singular Varieties ◽

Underlying Space ◽

And Topology ◽

Kähler Einstein Metrics

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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